We investigate phase transitions in boson-fermion systems. We propose an
analytically solvable model (E(5/12)) to describe odd nuclei at the critical
point in the transition from the spherical to $\gamma$-unstable behaviour. In
the model, a boson core described within the Bohr Hamiltonian interacts with an
unpaired particle assumed to be moving in the three single particle orbitals
j=1/2,3/2,5/2. Energy spectra and electromagnetic transitions at the critical
point compare well with the results obtained within the Interacting Boson
Fermion Model, with a boson-fermion Hamiltonian that describes the same
physical situation.Comment: Phys. Rev. Lett. (in press

A. Vitturi

C. E. Alonso

F. Iachello

J. M. Arias

English

arXiv

C. E.  Alonso

Crossref

Critical-Point Symmetries in Boson-Fermion Systems: The Case of Shape Transitions in Odd Nuclei in a Multiorbit Model

We investigate phase transitions in boson-fermion systems. We propose an analytically solvable model [E(5/12)] to describe odd nuclei at the critical point in the transition from the spherical to gamma-unstable behavior. In the model, a boson core described within the Bohr Hamiltonian interacts with an unpaired particle assumed to be moving in the three single-particle orbitals j = 1/2, 3/2, 5/2. Energy spectra and electromagnetic transitions at the critical point compare well with the results obtained within the interacting boson-fermion model, with a boson-fermion Hamiltonian that describes the same physical situation

ALONSO CE

ARIAS JM

VITTURI, ANDREA

Archivio istituzionale della ricerca - Università di Padova

Critical-Point Symmetries in Boson-Fermion Systems: The Case of Shape Transitionsin Odd Nuclei in a Multiorbit ModelC. E. Alonso,1 J. M. Arias,1 and A. Vitturi1,21Departamento de Fı́sica Atómica, Molecular y Nuclear, Facultad de Fı́sica, Universidad de Sevilla,Apartado 1065, 41080 Sevilla, Spain2Dipartimento di Fisica Galileo Galilei and INFN, Via Marzolo 8, 35131 Padova, Italy(Received 28 June 2006; revised manuscript received 29 November 2006; published 29 January 2007)We investigate phase transitions in boson-fermion systems. We propose an analytically solvable model[E5=12] to describe odd nuclei at the critical point in the transition from the spherical to -unstablebehavior. In the model, a boson core described within the Bohr Hamiltonian interacts with an unpairedparticle assumed to be moving in the three single-particle orbitals j  1=2, 3=2, 5=2. Energy spectra andelectromagnetic transitions at the critical point compare well with the results obtained within theinteracting boson-fermion model, with a boson-fermion Hamiltonian that describes the same physicalsituation.DOI: 10.1103/PhysRevLett.98.052501 PACS numbers: 21.60.Fw, 21.60.Ev, 27.80.+wThe occurrence of shape transitional behavior character-izes both even-even and odd nuclei. Because of the furthercomplexity embodied in the description of the coupling ofthe odd particle to the even core, formalisms describingphase transitions have been initially mainly devised foreven systems. Only recently the study of phase transitionsin odd nuclei has been revitalized. The first case consideredby Iachello [1–3] has been the case of a j  3=2 fermioncoupled to a boson core that undergoes a transition fromspherical to  unstable. At the critical point, an elegantanalytic solution, called E5=4, has been obtained startingfrom the Bohr Hamiltonian. The possible occurrence ofthis symmetry in 135Ba has been recently suggested [4].However, the restriction of the fermion space to a single-jorbit makes difficult the identification of a particular nu-cleus as critical.In this Letter we consider the richer case of a collectivecore coupled to a particle moving in the j  1=2, 3=2, and5=2 single-particle orbits, with the boson part undergoing atransition from sphericity to deformed  instability. Thissituation is likely to occur in the mass region around A 190, for example, in odd-neutron Pt where the relevantactive orbitals around the Fermi surface are the 3p1=2,3p3=2, and 2f5=2, in odd-proton Ir isotopes with activeorbits 2d5=2, 2d3=2, and 3s1=2, and also in the region aroundA  130, in odd-neutron Ba isotopes with relevant orbits2d5=2, 2d3=2, and 3s1=2, just to cite a few of them. At thecritical point we extend the E5=4 model to the multi-jcase, obtaining a new analytic solution, which we will callE5=12. Knowing the analytic form of the eigenfunctions,it is then easy to calculate spectroscopic observables, suchas electromagnetic transition probabilities. To check thevalidity of the model, we will also present more micro-scopic calculations based on the interacting boson-fermionmodel (IBFM) [5], with a choice of the boson-fermionHamiltonian that describes the same physical situation.We will show that the E5=12 and IBFM models givecomparable behaviors.We will then consider the coupling of a collective core,described within the quadrupole collective BohrHamiltonian, with an odd particle moving in the 1=2,3=2, 5=2 orbitals. One can use the separation of thesingle-particle angular momentum into a pseudospin andpseudo-orbital angular momentum (0 and 2) and assume nocoupling due to the pseudospin. The pseudo-orbital parttransforms as the representations (F1 , 0) of OF5, with F1being either 0 or 1 [6]. One can therefore use a modelHamiltonian of the form H  @22B14@@4@@12 sin3@@sin3@@144XQ2sin2 23 u kg2L̂B  L̂F  k0gL̂2F; (1)with u  0;  < w;u  1;  	 w;g @22B2;(2)where L̂B and L̂F are the five-dimensional boson andfermion orbital angular momenta, and the dot  indicatesthe five-dimensional scalar product. This Hamiltonian isthe generalization of the Hamiltonian used in the E5=4model. It should be noted that the last term was notincluded in E5=4. In that case a single-j was treatedand, consequently, a single OF5 representation was in-volved. For the E5=12 multi-j case two OF5 represen-tations are present and the last term in (1) is needed in orderPRL 98, 052501 (2007) P H Y S I C A L R E V I E W L E T T E R Sweek ending2 FEBRUARY 20070031-9007=07=98(5)=052501(4) 052501-1 © 2007 The American Physical Societyto include the energy difference between j  1=2 and j 3=2, 5=2 single-particle orbits. The choice of an infinitesquare well in the  variable should mimic the actualsituation at the critical point of the transition from thespherical situation (potential with a minimum in   0)to the deformed  instability (potential with a minimum fora finite value of ). In this case, since the group character-izing the core transition is the Euclidean one in five di-mensions E5 and the algebra for the fermion space isU12, we will name the model as E5=12.With this choice the total wave function can be factor-ized in the form   F; i; orb  spinJBF; (3)where orb and spin represent the particle coordinatesassociated with the pseudo-orbital and the pseudospinspaces. The wave function  is characterized by goodquantum numbers (1, 2, B1 , F1 , LBF,ML) and is given by 1;2;B1 ;F1 ;LBF;ML; i; orb X‘B;m‘B ;‘F;m‘FB1 ; 0 F1 ; 0 j 1; 2‘B ‘F j LBF‘B ‘F j LBFm‘B m‘F j ML	B1 ;0;‘B;m‘B ; iXF1 ;0;‘F;m‘F orb: (4)The function Xorb describes the five-dimensionalpseudo-orbital part, while spin is the pseudospinwave function, which does not contribute to the energy.The equation for F then acquires the familiar form @22B14@@4@@@22B2 uF  EF;(5)with   B1 B1  3  k11  3  22  1 B1 B1  3  F1 F1  3  k0F1 F1  3: (6)The solutions of this equation, as in the case of theE5=4 model, are given in terms of Bessel functions oforder   9=4p, in the form F;f1;2;B1 ;F1 g  c;f1;2;B1 ;F1 g3=2 Jx;f1;2;B1 ;F1 g=w; (7)where c;f1;2;B1 ;F1 g is a normalization constant andx;f1;2;B1 ;F1 g the th zero of Jz. The correspondingeigenvalues are given as E;f1;2;B1 ;F1 g @22Bx;f1;2;B1 ;F1 gw2: (8)The corresponding spectrum is given in the upper frameof Fig. 1 for the choice of the parameters k  1=4 andk0  5=2. Each state is characterized by the , B1 ,F1 1; 2 quantum numbers and arranged in bands accord-ing to the counting quantum number . The factorization ofthe pseudospin part give rise to repeated level couplets.Most of the states are therefore degenerate, starting fromthe first excited 3=2–5=2 doublet. The values of the angularmomenta (twice) for each degenerate state characterizedby the OBF5 boson-fermion (1, 2) labels are given inthe lowest inset panel. The lowest band with   1 comesfrom the coupling of the boson states (B1 , 0) with thefermion states (F1  0, 0). It is worth mentioning that allthese states with F1  0 have precisely the same energiesas the states in the even E5 model. The set of three bandsassociated to the first excited   1 structure comes fromthe coupling of (B1 , 0) with (F1  1, 0) which producesstates (1, 2) of the full boson-fermion system labeled as(B1  1, 0), (B1 , 1), and (B1  1, 0) [6]. In addition to these  1 bands, excited   2; 3; . . . structures appear.We can now calculate quadrupole transitions. In thiscase only the collective part of the quadrupole transitionoperator has been used, as in Refs. [1–3]. The calculationreduces to a single numerical integral over the  variable.In Fig. 1 we have chosen to represent the BE2’s betweenthe highest spins of initial and final multiplets [the BE2associated with the lowest 5=2 1=2 transition is normal-ized to 100], since most levels are multiply degenerate.Note that states with different F1 are not connected be-tween them by E2 transitions.To check the validity of the model we can describe asimilar physical situation within the framework of theIBFM [5], where the coupling of a single fermion to theeven-even boson core is described by the Hamiltonian H  HB HF  VBF; (9)where the term VBF couples the boson and fermion degreesof freedom. We remind that critical-point symmetries, asE5, X5, or E5=4, although inspired by models withinteracting bosons, can only be approximately obtainedwithin these models. However, the comparison with theIBM and IBFM turned out to be useful, e.g., in the case ofthe E5 and E5=4 symmetries, for a better understandingof the characteristics of these models [7–9]. In these cases,the Hamiltonian used for the bosonic part was a parame-trized one that produces a transition between spherical and-unstable shapes, of the form HB  xn̂d  1 x=NQ̂B  Q̂B; (10)PRL 98, 052501 (2007) P H Y S I C A L R E V I E W L E T T E R Sweek ending2 FEBRUARY 2007052501-2where the operators in the Hamiltonian are given by n̂d Pdyd and Q̂B  sy  ~d dy  ~s2, and N is thetotal number of bosons. This Hamiltonian can be recastinto the form HB  xC1UB5 1 x2NfC2OB6  C2OB5g;(11)where C1UB5 is the linear Casimir operator of theUB5 algebra, while C2OB6 and C2OB5 are thequadratic Casimir operators of the OB6 and OB5 alge-bras. By using the intrinsic frame formalism (see, forinstance, [10] ), one can show that the critical point occursat xcrit 4N85N8 .In analogy with this boson Hamiltonian, which has beenextensively studied is Refs. [7,11,12], one could choose[13] in our case a boson-fermion Hamiltonian of the form H  xn̂d  n̂3=2  n̂5=2 1 xNQ̂BF  Q̂BF xC1UBF5 1 x2NfC2OBF6  C2OBF5g;(12)which implies a certain choice of the total quadrupoleoperator and quasiparticle energies [5]. In particular, it isworth noting that the j  3=2 and 5=2 orbits are degener-ate and higher in energy than the j  1=2 orbit. Such aHamiltonian has been used thoroughly in the literature and,in particular, in a previous study of shape phase transitionin odd-mass nuclei using a supersymmetric approach [14].It has the advantage that one recovers, in the extreme cases,the Bose-Fermi symmetry [5] associated with OBF6 UFs 2 (for x  0) and the vibrational UBF5 UFs 2 (forx  1). On the other hand, such an IBFM Hamiltonian onlyapproximately preserves [13] the B1 and F1 quantum num-bers, while those are good quantum numbers for E5=12.Although it will be possible to establish anyway a corre-spondence between E5=12 and IBFM states, selectionrules for electromagnetic and one-particle transfer transi-tions will be different.For a better comparison with the E5=12 results wehave therefore preferred to use here at the critical point aboson-fermion Hamiltonian of the form H  xC1UB5 1 x2NfC2OB6  C2OB5gk2NfC2OBF5  C2OB5  C2OF5gk02NC2OF5; (13)where C2OF5, C2OBF5 are the quadratic Casimiroperators of the OF5 and OBF5 algebras. ThisHamiltonian is designed to mimic as much as possiblethe corresponding Hamiltonian in E5=12, since theterms 2L̂B  L̂F and L̂2F in (1) can be written in terms ofCasimir operators as 2L̂B  L̂F  1=2fC2OBF5 C2OB5  C2OF5g and L̂2F  1=2C2OF5.This boson-fermion Hamiltonian preserves by constructionthe B1 and F1 quantum numbers and all states are thereforecharacterized by the same quantum numbers and degener-acies as in the E5=12 case. For a better clarification, weshow in Fig. 2 the lattice of algebras relevant for ourproblem.The corresponding spectrum is given in the lower frameof Fig. 1. A number N  7 of bosons is assumed as atypical value. Other choices of N do not alter the qualita-tive behavior of our results. With the choice N  7 thecritical point occurs at xcrit 4N85N8 2027  0:74. In theHamiltonian the values for k and k0 have been assumedto be equal to the corresponding values in the E5=12 casepresented in the upper frame. FIG. 1. Upper frame: energy levels in the odd system at theE5=12 critical point (normalized to the energy of the firstexcited state). Each (degenerate) state is characterized by the, B1 , F1 , (1, 2) quantum numbers. The values k  1=4,k0  5=2 have been used in the Hamiltonian (1). The values of(twice) the angular momenta for each degenerate state can beextracted from the inset in the lowest panel. Some selectedvalues of BE2’s, corresponding to the transitions between thehighest spins of initial and final multiplets, are given in the figure[the BE2 associated with the lowest 5=2 1=2 transition isnormalized to 100]. Lower frame: energy levels in the oddsystem within the IBFM Hamiltonian (13) at the critical point.A number of bosons N  7 has been assumed. The values of kand k0 are the same as in the E5=12 Hamiltonian. For theBE2’s, see the upper frame.PRL 98, 052501 (2007) P H Y S I C A L R E V I E W L E T T E R Sweek ending2 FEBRUARY 2007052501-3The spectrum has a structure that clearly resembles theone obtained within the E5=12 model (upper frame).Energies of selected states in the two models are comparedin Table I. As noted before, in the E5=12 model all stateswith F1  0 (i.e., with the odd particle in the j  1=2state) have precisely the same energies as the states inthe even E5 model. Similarly, in the IBFM these stateshave the same energy of the corresponding states in theboson Hamiltonian (10). The agreement for the energies ofthese states in E5=12 and IBFM is therefore that alreadydiscussed in Ref. [7] for the bosonic part. Excited bandswith F1  1 (i.e., with the odd particle in the j  3=2, 5=2states), on the other hand, seem to be somewhat com-pressed in E5=12 with respect to the IBFM.The same kind of agreement is also present in theelectromagnetic transitions. Consistently with the choicepresented in the E5=12, we assume in the IBFM as E2transition operator just the boson (collective) quadrupoleoperator, i.e., T̂E2  Q̂B. The order of magnitude of thedifferent BE2 values are reflected in different thicknessand styles of the arrows shown in Fig. 1. Given the choiceof the IBFM Hamiltonian (13) the same selection rulesvalid in the E5=12model also apply to the (B1 , F1 , 1, 2)quantum numbers of initial and final states in the IBFMapproach. The BE2 values of selected transitions areexplicitly given in the figure. All BE2 values are normal-ized to 100 for the first 5=2–1=2 transition. Similarly towhat happens for the energies of the levels, the overallbehavior is very similar to that of the E5=12.In this Letter we have presented an analytic model[E5=12] for the critical point of the phase transitionfrom spherical to -unstable shapes in odd nuclei, i.e., amixture of fermion and boson degrees of freedom. In themodel the fermion is allowed to occupy many single-particle orbitals. Spectrum and intensities of electromag-netic transitions have been compared with those obtainedwithin the interacting boson-fermion model showing com-parable behaviors. We think that our model, which includesthe more realistic many j-orbits case compared with theE5=4 model, should offer more chances for the occur-rence of critical-point symmetries in odd nuclei, in addi-tion to those already found in the case of even nuclei.This work has been partially supported by the SpanishMinisterio de Educación y Ciencia and by the Europeanregional development fund (FEDER) under ProjectNo. FIS2005-01105, and by INFN. A. V. acknowledgesfinancial support from SEUI (Spanish Ministerio deEducación y Ciencia) for a sabbatical year at Universityof Sevilla.[1] F. Iachello, Phys. Rev. Lett. 95, 052503 (2005).[2] F. Iachello, in Symmetries and Low-Energy PhaseTransitions in Nuclear Structure Physics, edited by G.Lo Bianco (Universitá degli Studi di Camerino,Camerino, 2005).[3] M. A. Caprio and F. Iachello, Nucl. Phys. A781, 26(2007).[4] M. S. Fetea et al., Phys. Rev. C 73, 051301(R) (2006).[5] F. Iachello and P. Van Isacker, The Interacting BosonFermion Model (Cambridge University Press,Cambridge, England, 1991).[6] P. Van Isacker, A. Frank, and H.-Z. Sun, Ann. Phys. (N.Y.)157, 183 (1984).[7] J. M. Arias, C. E. Alonso, A. Vitturi, J. E. Garcı́a-Ramos,J. Dukelsky, and A. Frank, Phys. Rev. C 68, 041302(R)(2003).[8] C. E. Alonso, J. M. Arias, L. Fortunato, and A. Vitturi,Phys. Rev. C 72, 061302(R) (2005).[9] C. E. Alonso, J. M. Arias, and A. Vitturi, Phys. Rev. C 74,027301 (2006).[10] J. Jolie, P. Cejnar, R. F. Casten, S. Heinze, A. Linnemann,and V. Werner, Phys. Rev. Lett. 89, 182502 (2002).[11] F. Iachello, Phys. Rev. Lett. 85, 3580 (2000).[12] P. S. Turner and D. J. Rowe, Nucl. Phys. A756, 333 (2005).[13] C. E. Alonso et al. (to be published).[14] J. Jolie, S. Heinze, P. Van Isacker, and R. F. Casten, Phys.Rev. C 70, 011305 (2004).TABLE I. Energies of selected states in E5=12 and IBFM(normalized to the first excited multiplet). The values k  1=4,k0  5=2 have been used in both Hamiltonians (1) and (13). Inthe IBFM calculation, a number N  7 of bosons has been used.Level , B1 , F1 1; 2 Energy E5=12 Energy IBFM1,1,0 (1,0) 1.00 1.001,2,0 (2,0) 2.20 2.302,0,0 (0,0) 3.03 2.861,0,1 (1,0) 2.20 2.691,1,1 (1,1) 3.00 3.82 UB(6) UF(12)OB(6)OB(5)OBF(5)OFk(6) UFs(2)OFk(5)OBF(3) SUFs(2)SpinBF(3)⊗⊗UB(5)FIG. 2 (color online). The lattice of algebras involved in theIBFM Hamiltonian (13).PRL 98, 052501 (2007) P H Y S I C A L R E V I E W L E T T E R Sweek ending2 FEBRUARY 2007052501-4

We investigate phase transitions in boson-fermion systems. We propose an analytically solvable model [E(5/12)] to describe odd nuclei at the critical point in the transition from the spherical to γ-unstable behavior. In the model, a boson core described within the Bohr Hamiltonian interacts with an unpaired particle assumed to be moving in the three single-particle orbitals j=1/2, 3/2, 5/2. Energy spectra and electromagnetic transitions at the critical point compare well with the results obtained within the interacting boson-fermion model, with a boson-fermion Hamiltonian that describes the same physical situation.Ministerio  de  Educación y Ciencia y FEDER FIS2005-0110

Alonso Alonso, Clara Eugenia

Arias Carrasco, José Miguel

Vitturi, Andrea

idUS. Depósito de Investigación Universidad de Sevilla

Critical-Point Symmetries in Boson-Fermion Systems: The Case of Shape Transitionsin Odd Nuclei in a Multiorbit ModelC. E. Alonso,1 J. M. Arias,1 and A. Vitturi1,21Departamento de Fı́sica Atómica, Molecular y Nuclear, Facultad de Fı́sica, Universidad de Sevilla,Apartado 1065, 41080 Sevilla, Spain2Dipartimento di Fisica Galileo Galilei and INFN, Via Marzolo 8, 35131 Padova, Italy(Received 28 June 2006; revised manuscript received 29 November 2006; published 29 January 2007)We investigate phase transitions in boson-fermion systems. We propose an analytically solvable model[E5=12] to describe odd nuclei at the critical point in the transition from the spherical to -unstablebehavior. In the model, a boson core described within the Bohr Hamiltonian interacts with an unpairedparticle assumed to be moving in the three single-particle orbitals j  1=2, 3=2, 5=2. Energy spectra andelectromagnetic transitions at the critical point compare well with the results obtained within theinteracting boson-fermion model, with a boson-fermion Hamiltonian that describes the same physicalsituation.DOI: 10.1103/PhysRevLett.98.052501 PACS numbers: 21.60.Fw, 21.60.Ev, 27.80.+wThe occurrence of shape transitional behavior character-izes both even-even and odd nuclei. Because of the furthercomplexity embodied in the description of the coupling ofthe odd particle to the even core, formalisms describingphase transitions have been initially mainly devised foreven systems. Only recently the study of phase transitionsin odd nuclei has been revitalized. The first case consideredby Iachello [1–3] has been the case of a j  3=2 fermioncoupled to a boson core that undergoes a transition fromspherical to  unstable. At the critical point, an elegantanalytic solution, called E5=4, has been obtained startingfrom the Bohr Hamiltonian. The possible occurrence ofthis symmetry in 135Ba has been recently suggested [4].However, the restriction of the fermion space to a single-jorbit makes difficult the identification of a particular nu-cleus as critical.In this Letter we consider the richer case of a collectivecore coupled to a particle moving in the j  1=2, 3=2, and5=2 single-particle orbits, with the boson part undergoing atransition from sphericity to deformed  instability. Thissituation is likely to occur in the mass region around A 190, for example, in odd-neutron Pt where the relevantactive orbitals around the Fermi surface are the 3p1=2,3p3=2, and 2f5=2, in odd-proton Ir isotopes with activeorbits 2d5=2, 2d3=2, and 3s1=2, and also in the region aroundA  130, in odd-neutron Ba isotopes with relevant orbits2d5=2, 2d3=2, and 3s1=2, just to cite a few of them. At thecritical point we extend the E5=4 model to the multi-jcase, obtaining a new analytic solution, which we will callE5=12. Knowing the analytic form of the eigenfunctions,it is then easy to calculate spectroscopic observables, suchas electromagnetic transition probabilities. To check thevalidity of the model, we will also present more micro-scopic calculations based on the interacting boson-fermionmodel (IBFM) [5], with a choice of the boson-fermionHamiltonian that describes the same physical situation.We will show that the E5=12 and IBFM models givecomparable behaviors.We will then consider the coupling of a collective core,described within the quadrupole collective BohrHamiltonian, with an odd particle moving in the 1=2,3=2, 5=2 orbitals. One can use the separation of thesingle-particle angular momentum into a pseudospin andpseudo-orbital angular momentum (0 and 2) and assume nocoupling due to the pseudospin. The pseudo-orbital parttransforms as the representations (F1 , 0) of OF5, with F1being either 0 or 1 [6]. One can therefore use a modelHamiltonian of the form H  @22B14@@4@@12 sin3@@sin3@@144XQ2sin2 23 u kg2L̂B  L̂F  k0gL̂2F; (1)with u  0;  < w;u  1;  	 w;g @22B2;(2)where L̂B and L̂F are the five-dimensional boson andfermion orbital angular momenta, and the dot  indicatesthe five-dimensional scalar product. This Hamiltonian isthe generalization of the Hamiltonian used in the E5=4model. It should be noted that the last term was notincluded in E5=4. In that case a single-j was treatedand, consequently, a single OF5 representation was in-volved. For the E5=12 multi-j case two OF5 represen-tations are present and the last term in (1) is needed in orderPRL 98, 052501 (2007) P H Y S I C A L R E V I E W L E T T E R Sweek ending2 FEBRUARY 20070031-9007=07=98(5)=052501(4) 052501-1 © 2007 The American Physical Societybrought to you by COREView metadata, citation and similar papers at core.ac.ukprovided by idUS. Depósito de Investigación Universidad de Sevillato include the energy difference between j  1=2 and j 3=2, 5=2 single-particle orbits. The choice of an infinitesquare well in the  variable should mimic the actualsituation at the critical point of the transition from thespherical situation (potential with a minimum in   0)to the deformed  instability (potential with a minimum fora finite value of ). In this case, since the group character-izing the core transition is the Euclidean one in five di-mensions E5 and the algebra for the fermion space isU12, we will name the model as E5=12.With this choice the total wave function can be factor-ized in the form   F; i; orb  spinJBF; (3)where orb and spin represent the particle coordinatesassociated with the pseudo-orbital and the pseudospinspaces. The wave function  is characterized by goodquantum numbers (1, 2, B1 , F1 , LBF,ML) and is given by 1;2;B1 ;F1 ;LBF;ML; i; orb X‘B;m‘B ;‘F;m‘FB1 ; 0 F1 ; 0 j 1; 2‘B ‘F j LBF‘B ‘F j LBFm‘B m‘F j ML	B1 ;0;‘B;m‘B ; iXF1 ;0;‘F;m‘F orb: (4)The function Xorb describes the five-dimensionalpseudo-orbital part, while spin is the pseudospinwave function, which does not contribute to the energy.The equation for F then acquires the familiar form @22B14@@4@@@22B2 uF  EF;(5)with   B1 B1  3  k11  3  22  1 B1 B1  3  F1 F1  3  k0F1 F1  3: (6)The solutions of this equation, as in the case of theE5=4 model, are given in terms of Bessel functions oforder   9=4p, in the form F;f1;2;B1 ;F1 g  c;f1;2;B1 ;F1 g3=2 Jx;f1;2;B1 ;F1 g=w; (7)where c;f1;2;B1 ;F1 g is a normalization constant andx;f1;2;B1 ;F1 g the th zero of Jz. The correspondingeigenvalues are given as E;f1;2;B1 ;F1 g @22Bx;f1;2;B1 ;F1 gw2: (8)The corresponding spectrum is given in the upper frameof Fig. 1 for the choice of the parameters k  1=4 andk0  5=2. Each state is characterized by the , B1 ,F1 1; 2 quantum numbers and arranged in bands accord-ing to the counting quantum number . The factorization ofthe pseudospin part give rise to repeated level couplets.Most of the states are therefore degenerate, starting fromthe first excited 3=2–5=2 doublet. The values of the angularmomenta (twice) for each degenerate state characterizedby the OBF5 boson-fermion (1, 2) labels are given inthe lowest inset panel. The lowest band with   1 comesfrom the coupling of the boson states (B1 , 0) with thefermion states (F1  0, 0). It is worth mentioning that allthese states with F1  0 have precisely the same energiesas the states in the even E5 model. The set of three bandsassociated to the first excited   1 structure comes fromthe coupling of (B1 , 0) with (F1  1, 0) which producesstates (1, 2) of the full boson-fermion system labeled as(B1  1, 0), (B1 , 1), and (B1  1, 0) [6]. In addition to these  1 bands, excited   2; 3; . . . structures appear.We can now calculate quadrupole transitions. In thiscase only the collective part of the quadrupole transitionoperator has been used, as in Refs. [1–3]. The calculationreduces to a single numerical integral over the  variable.In Fig. 1 we have chosen to represent the BE2’s betweenthe highest spins of initial and final multiplets [the BE2associated with the lowest 5=2 1=2 transition is normal-ized to 100], since most levels are multiply degenerate.Note that states with different F1 are not connected be-tween them by E2 transitions.To check the validity of the model we can describe asimilar physical situation within the framework of theIBFM [5], where the coupling of a single fermion to theeven-even boson core is described by the Hamiltonian H  HB HF  VBF; (9)where the term VBF couples the boson and fermion degreesof freedom. We remind that critical-point symmetries, asE5, X5, or E5=4, although inspired by models withinteracting bosons, can only be approximately obtainedwithin these models. However, the comparison with theIBM and IBFM turned out to be useful, e.g., in the case ofthe E5 and E5=4 symmetries, for a better understandingof the characteristics of these models [7–9]. In these cases,the Hamiltonian used for the bosonic part was a parame-trized one that produces a transition between spherical and-unstable shapes, of the form HB  xn̂d  1 x=NQ̂B  Q̂B; (10)PRL 98, 052501 (2007) P H Y S I C A L R E V I E W L E T T E R Sweek ending2 FEBRUARY 2007052501-2where the operators in the Hamiltonian are given by n̂d Pdyd and Q̂B  sy  ~d dy  ~s2, and N is thetotal number of bosons. This Hamiltonian can be recastinto the form HB  xC1UB5 1 x2NfC2OB6  C2OB5g;(11)where C1UB5 is the linear Casimir operator of theUB5 algebra, while C2OB6 and C2OB5 are thequadratic Casimir operators of the OB6 and OB5 alge-bras. By using the intrinsic frame formalism (see, forinstance, [10] ), one can show that the critical point occursat xcrit 4N85N8 .In analogy with this boson Hamiltonian, which has beenextensively studied is Refs. [7,11,12], one could choose[13] in our case a boson-fermion Hamiltonian of the form H  xn̂d  n̂3=2  n̂5=2 1 xNQ̂BF  Q̂BF xC1UBF5 1 x2NfC2OBF6  C2OBF5g;(12)which implies a certain choice of the total quadrupoleoperator and quasiparticle energies [5]. In particular, it isworth noting that the j  3=2 and 5=2 orbits are degener-ate and higher in energy than the j  1=2 orbit. Such aHamiltonian has been used thoroughly in the literature and,in particular, in a previous study of shape phase transitionin odd-mass nuclei using a supersymmetric approach [14].It has the advantage that one recovers, in the extreme cases,the Bose-Fermi symmetry [5] associated with OBF6 UFs 2 (for x  0) and the vibrational UBF5 UFs 2 (forx  1). On the other hand, such an IBFM Hamiltonian onlyapproximately preserves [13] the B1 and F1 quantum num-bers, while those are good quantum numbers for E5=12.Although it will be possible to establish anyway a corre-spondence between E5=12 and IBFM states, selectionrules for electromagnetic and one-particle transfer transi-tions will be different.For a better comparison with the E5=12 results wehave therefore preferred to use here at the critical point aboson-fermion Hamiltonian of the form H  xC1UB5 1 x2NfC2OB6  C2OB5gk2NfC2OBF5  C2OB5  C2OF5gk02NC2OF5; (13)where C2OF5, C2OBF5 are the quadratic Casimiroperators of the OF5 and OBF5 algebras. ThisHamiltonian is designed to mimic as much as possiblethe corresponding Hamiltonian in E5=12, since theterms 2L̂B  L̂F and L̂2F in (1) can be written in terms ofCasimir operators as 2L̂B  L̂F  1=2fC2OBF5 C2OB5  C2OF5g and L̂2F  1=2C2OF5.This boson-fermion Hamiltonian preserves by constructionthe B1 and F1 quantum numbers and all states are thereforecharacterized by the same quantum numbers and degener-acies as in the E5=12 case. For a better clarification, weshow in Fig. 2 the lattice of algebras relevant for ourproblem.The corresponding spectrum is given in the lower frameof Fig. 1. A number N  7 of bosons is assumed as atypical value. Other choices of N do not alter the qualita-tive behavior of our results. With the choice N  7 thecritical point occurs at xcrit 4N85N8 2027  0:74. In theHamiltonian the values for k and k0 have been assumedto be equal to the corresponding values in the E5=12 casepresented in the upper frame. FIG. 1. Upper frame: energy levels in the odd system at theE5=12 critical point (normalized to the energy of the firstexcited state). Each (degenerate) state is characterized by the, B1 , F1 , (1, 2) quantum numbers. The values k  1=4,k0  5=2 have been used in the Hamiltonian (1). The values of(twice) the angular momenta for each degenerate state can beextracted from the inset in the lowest panel. Some selectedvalues of BE2’s, corresponding to the transitions between thehighest spins of initial and final multiplets, are given in the figure[the BE2 associated with the lowest 5=2 1=2 transition isnormalized to 100]. Lower frame: energy levels in the oddsystem within the IBFM Hamiltonian (13) at the critical point.A number of bosons N  7 has been assumed. The values of kand k0 are the same as in the E5=12 Hamiltonian. For theBE2’s, see the upper frame.PRL 98, 052501 (2007) P H Y S I C A L R E V I E W L E T T E R Sweek ending2 FEBRUARY 2007052501-3The spectrum has a structure that clearly resembles theone obtained within the E5=12 model (upper frame).Energies of selected states in the two models are comparedin Table I. As noted before, in the E5=12 model all stateswith F1  0 (i.e., with the odd particle in the j  1=2state) have precisely the same energies as the states inthe even E5 model. Similarly, in the IBFM these stateshave the same energy of the corresponding states in theboson Hamiltonian (10). The agreement for the energies ofthese states in E5=12 and IBFM is therefore that alreadydiscussed in Ref. [7] for the bosonic part. Excited bandswith F1  1 (i.e., with the odd particle in the j  3=2, 5=2states), on the other hand, seem to be somewhat com-pressed in E5=12 with respect to the IBFM.The same kind of agreement is also present in theelectromagnetic transitions. Consistently with the choicepresented in the E5=12, we assume in the IBFM as E2transition operator just the boson (collective) quadrupoleoperator, i.e., T̂E2  Q̂B. The order of magnitude of thedifferent BE2 values are reflected in different thicknessand styles of the arrows shown in Fig. 1. Given the choiceof the IBFM Hamiltonian (13) the same selection rulesvalid in the E5=12model also apply to the (B1 , F1 , 1, 2)quantum numbers of initial and final states in the IBFMapproach. The BE2 values of selected transitions areexplicitly given in the figure. All BE2 values are normal-ized to 100 for the first 5=2–1=2 transition. Similarly towhat happens for the energies of the levels, the overallbehavior is very similar to that of the E5=12.In this Letter we have presented an analytic model[E5=12] for the critical point of the phase transitionfrom spherical to -unstable shapes in odd nuclei, i.e., amixture of fermion and boson degrees of freedom. In themodel the fermion is allowed to occupy many single-particle orbitals. Spectrum and intensities of electromag-netic transitions have been compared with those obtainedwithin the interacting boson-fermion model showing com-parable behaviors. We think that our model, which includesthe more realistic many j-orbits case compared with theE5=4 model, should offer more chances for the occur-rence of critical-point symmetries in odd nuclei, in addi-tion to those already found in the case of even nuclei.This work has been partially supported by the SpanishMinisterio de Educación y Ciencia and by the Europeanregional development fund (FEDER) under ProjectNo. FIS2005-01105, and by INFN. A. V. acknowledgesfinancial support from SEUI (Spanish Ministerio deEducación y Ciencia) for a sabbatical year at Universityof Sevilla.[1] F. Iachello, Phys. Rev. Lett. 95, 052503 (2005).[2] F. Iachello, in Symmetries and Low-Energy PhaseTransitions in Nuclear Structure Physics, edited by G.Lo Bianco (Universitá degli Studi di Camerino,Camerino, 2005).[3] M. A. Caprio and F. Iachello, Nucl. Phys. A781, 26(2007).[4] M. S. Fetea et al., Phys. Rev. C 73, 051301(R) (2006).[5] F. Iachello and P. Van Isacker, The Interacting BosonFermion Model (Cambridge University Press,Cambridge, England, 1991).[6] P. Van Isacker, A. Frank, and H.-Z. Sun, Ann. Phys. (N.Y.)157, 183 (1984).[7] J. M. Arias, C. E. Alonso, A. Vitturi, J. E. Garcı́a-Ramos,J. Dukelsky, and A. Frank, Phys. Rev. C 68, 041302(R)(2003).[8] C. E. Alonso, J. M. Arias, L. Fortunato, and A. Vitturi,Phys. Rev. C 72, 061302(R) (2005).[9] C. E. Alonso, J. M. Arias, and A. Vitturi, Phys. Rev. C 74,027301 (2006).[10] J. Jolie, P. Cejnar, R. F. Casten, S. Heinze, A. Linnemann,and V. Werner, Phys. Rev. Lett. 89, 182502 (2002).[11] F. Iachello, Phys. Rev. Lett. 85, 3580 (2000).[12] P. S. Turner and D. J. Rowe, Nucl. Phys. A756, 333 (2005).[13] C. E. Alonso et al. (to be published).[14] J. Jolie, S. Heinze, P. Van Isacker, and R. F. Casten, Phys.Rev. C 70, 011305 (2004).TABLE I. Energies of selected states in E5=12 and IBFM(normalized to the first excited multiplet). The values k  1=4,k0  5=2 have been used in both Hamiltonians (1) and (13). Inthe IBFM calculation, a number N  7 of bosons has been used.Level , B1 , F1 1; 2 Energy E5=12 Energy IBFM1,1,0 (1,0) 1.00 1.001,2,0 (2,0) 2.20 2.302,0,0 (0,0) 3.03 2.861,0,1 (1,0) 2.20 2.691,1,1 (1,1) 3.00 3.82 UB(6) UF(12)OB(6)OB(5)OBF(5)OFk(6) UFs(2)OFk(5)OBF(3) SUFs(2)SpinBF(3)⊗⊗UB(5)FIG. 2 (color online). The lattice of algebras involved in theIBFM Hamiltonian (13).PRL 98, 052501 (2007) P H Y S I C A L R E V I E W L E T T E R Sweek ending2 FEBRUARY 2007052501-4

Spanish

Critical-point symmetries in Boson-Fermion systems: The case of shape transitions in odd nuclei in a multiorbit model

https://idus.us.es/bitstream/11441/62635/1/Critical-Point%20Symmetries%20in%20Boson-Fermion%20Systems%20The%20Case%20of%20Shape%20Transitions%20in%20Odd%20Nuclei%20in%20a%20Multiorbit%20Model.pdf

Critical point symmetries in boson-fermion systems. The case of shape transition in odd nuclei in a multi-orbit model

Abstract

Similar works

Full text

Available Versions

Crossref

Archivio istituzionale della ricerca - Università di Padova

idUS. Depósito de Investigación Universidad de Sevilla